Abstract
The H∞ problem for a nonlinear system is considered. The corresponding dynamic programming equation is a fully nonlinear, first-order, steady-state partial differential equation (PDE). The computation of the solution of a nonlinear, steady-state, first-order PDE is typically quite difficult. We consider an entirely new class of methods for the obtaining the solution of such PDEs. These methods are based on the linearity of the associated semi-group over the max-plus algebra. In particular, solution of the PDE is reduced to solution of a max-plus eigenvector problem for known unique eigenvalue 0. We consider the error analysis for such an algorithm. The errors are due to both the truncation of the basis expansion and computation of the matrix whose eigenvector one computes.
Research partially supported by NSF grant DMS-9971546.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
T. Basar and P. Bernhard, H∞-Optimal Control and Related Minimax Design Problems, Birkhäuser (1991).
J. A. Ball and J. W. Helton, H∞ control for nonlinear plants: connections with differential games, Proc. 28th IEEE Conf. Dec. Control, Tampa FL (1989), 956–962.
F. L. Baccelli, G. Cohen, G.J. Olsder and J.-P. Quadrat, Synchronization and Linearity, John Wiley (1992).
W. H. Fleming and W. M. McEneaney, A max-plus based algorithm for an HJB equation of nonlinear filtering, SIAM J. Control and Optim., 38 (2000), pp. 683–710.
W. H. Fleming, Deterministic nonlinear filtering, Annali Scuola Normale Superiore Pisa, Cl. Scienze Fisiche e Matematiche, Ser. IV, 25 (1997), 435–454.
E. Gallestey, M. R. James and W. M. McEneaney, Max-plus approximation methods in partially observed H∞ control, 38th IEEE Conf. on Decision and Control, 3011–3016.
M. R. James, A partial differential inequality for dissipative nonlinear systems, Systems and Control Letters, 21 (1993) 315–320.
V. P. Maslov, On a new principle of superposition for optimization problems, Russian Math. Surveys, 42 (1987) 43–54.
W. M. McEneaney, Max-Plus Eigenvector Representations for Solution of Nonlinear H∞ Problems: Basic Concepts, Submitted to IEEE Trans. Auto. Control.
W. M. McEneaney, Error Analysis of a Max-Plus Algorithm for a First-Order HJB Equation, Proc. Workshop On Max-Plus Algebras and Their Applications to Discrete-event Systems, Theoretical Computer Science, and Optimization, Prague 27–29 August 2001.
W. M. McEneaney, Convergence and error analysis for a max-plus algorithm, Proc. 39th IEEE Conf. on Decision and Control (2000), 1194–1199.
W. M. McEneaney and M. Horton, Max-plus eigenvector representations for nonlinear H∞ value functions, 37th IEEE Conf. on Decision and Control (1998), 3506–3511.
W. M. McEneaney, Robust/H∞ filtering for nonlinear systems, Systems and Control Letters, Vol. 33 (1998), 315–325.
W. M. McEneaney, A Uniqueness result for the Isaacs equation corresponding to nonlinear H∞ control, Math. Controls, Signals and Systems, Vol. 11 (1998), 303–334.
P. Soravia, H∞ control of nonlinear systems: differential games and viscosity solutions, SIAM J. Control and Optim., 34 (1996), 1071–1097.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Additional information
This paper is dedicated to Prof. Tyrone E. Duncan
Rights and permissions
Copyright information
© 2002 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
McEneaney, W.M. (2002). Error Analysis of a Max-Plus Algorithm for a First-Order HJB Equation. In: Pasik-Duncan, B. (eds) Stochastic Theory and Control. Lecture Notes in Control and Information Sciences, vol 280. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-48022-6_23
Download citation
DOI: https://doi.org/10.1007/3-540-48022-6_23
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-43777-2
Online ISBN: 978-3-540-48022-8
eBook Packages: Springer Book Archive